Little insights

Qiaochu has a post up on “going beyond your comfort zone” in mathematics. Towards the end of it, he lists many of the most commonly given good reasons for learning new areas of mathematics — and all of them are very good reasons. But there’s one that I want to talk about now which I don’t think I’ve seen mentioned before; it’s not so much a pedagogical principle as an autodidactical one.

Mathematical research is largely driven by insights — seeing that trick X can be applied to problem Y, for instance, or formulating conjecture C based on an interesting similarity between areas P and Q. Of course, top-quality work (especially theory-building work) is often focused around “big” insights, like the Langlands conjectures, or Serre and Grothendieck’s radical restructuring of algebraic geometry, or almost everything Riemann ever wrote. But as anyone who’s ever done serious mathematical research knows, those “big” insights are virtually always the culmination of a series of “small” insights that grew progressively larger, backed up by calculation or experimenting with special cases.

When you learn new mathematics, you get to have these “little insights” in spades — because the foundational level of any theory is built on them, and it’s much more fun to try to anticipate them in advance than to sit back and learn the material passively. Examples of these “little insights” might include:

If you know that a vector space is in a sense “an abelian group over a field,” and you’ve learned the definition of a ring, you might start thinking about the properties of “a vector space over a ring.” Congratulations — you’ve rediscovered module theory.

The realization that group presentations really just specify quotient groups of the free group — and the subsequent realization that every group is a quotient group of the free group on some number of generators.

The realization that group representations and their characters can tell us a lot about the structure of the group, including, often, whether it is simple or solvable.

The realization that powers don’t matter when we’re considering the solution set of a system of polynomials. (This leads directly to the Nullstellensatz.)

Noticing that every group action is continuous under the discrete topology.

Various and sundry other “hey, I’ve seen this before!” moments, for instance: seeing that taking the radical of an ideal is in many ways formally similar to taking the topological closure of a set; seeing that tensor products and/or direct sums are “essentially the same thing”; and many of the lower-level tricks in combinatorics that pop up time and again (e.g., counting mod 2).

I believe that every math course above perhaps freshman calculus should be taught in such a way as to maximize the number of these “aha!” moments where these little insights are gained, and something that’s taught in the next chapter or the chapter after that is anticipated, even if very informally. The insight muscle is among the mathematician’s most powerful tool, and we should exercise it early and often.

7 Responses to “Little insights”

You probably did this intentionally, but most of those little insights find their appropriate generalization in category theory. So I guess what you’re saying is that professors should teach with category theory in mind!

Module theory is an interesting example. Before I took 18.702, I thought that module theory was essentially the same as vector space theory. It’s not until one realizes that modules over a ring tells you things about the structure of the ring that module theory justifies itself as an independent course of study.

No, it wasn’t intentional; I think it mostly comes from the fact that most of what I’ve been studying recently(algebra, geometry) is appropriately generalized over category theory. (And actually, the usefulness of character theory and the “combinatorial hall of fame” don’t have good categorical explanation as far as I know — which is a big part of why I put them in!)

Module theory is definitely the deepest of the above examples, and I don’t think that it is easy to justify it as something worth studying without a push in the right direction. But it’s certainly worthwhile to start considering it, and realizing that it’s very different than linear algebra, as soon as you can!

Hi Harrison, please excuse the nitpick, but in “the subsequent realization that every group is a subgroup of the free group on some number of generators” did you mean to say quotient group instead of subgroup?

There’s a nice result (pretty nontrivial to prove using purely algebraic methods) that subgroups of free groups are themselves free.

I’d have to agree with this post, especially as I slowly understood that these so-called “universal properties” (which kept reappearing countless times, as tensor products, Grothendieck groups, free modules, etc.) could be phrased more generally in terms of category theory, especially in the language of representable functors. Then, all the uniqueness up-to-isomorphism follows from Yoneda, without any additional argument needed (although such an argument would be fairly routine anyway).

Incidentally, is that what you meant by saying “seeing that tensor products and/or direct sums are “essentially the same thing””?